In addition to conducting our analysis with biomass as the measurement of ecosystem function, in this document we report our results using net primary productivity (NPP).
Mirroring the manuscript’s central analysis, our models for the across-treatment effect were encoded as: Productivity ~ -1 + Stage + Stage:Shannon, and the within-treatment effect was encoded as: Productivity ~ -1 + Richness:Stage + Richness:Stage:Shannon. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.
The relationship between Shannon diversity and productivity was qualitatively similar to that of Shannon diversity for the majority of the models (4/6).
In Forest2, while the within-treatment slopes are consistent between biomass and productivity, the across-treatment slopes differ. During the with stage, the across-treatment slope is insignificant with biomass, but significant and slightly positive for productivity. During the without seed inflow phase, the significant and negative across-treatment slope of the relationship between biomass and Shannon diversity becomes insignificant for productivity.
Dryland displays the most variation between the two measures of ecosystem functioning. The predominant difference is shows in the within-treatment slopes, as they flip from being significantly positive to significantly negative in both the with and without seed inflow phases. Secondly, while the across-treatment slope is insignificant during the without seed inflow phase for biomass as the ecosystem functioning, for productivity the slope is significant and positive. The reason for this change is clerical, because while seed biomass is incorporated into the productivity calculations, it is left absent from the total biomass calculations.
Considering the relationship between our measure of the internal coexistence processes within each model and the across-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.
Considering the relationship between our measure of the internal coexistence processes within each model and the within-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.
This section of the document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and productivity as the focal ecosystem function.
Important terms:
Stage: With seed inflow, without seed inflowNinitial: Planted species richnessClark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 2.36 3.63 -4.71 9.64 1.00 2362 2294
## StageWithoutseedinflow -12.23 3.95 -19.90 -4.60 1.00 1903 1847
## StageWithseedinflow:Shannon 17.76 1.52 14.76 20.78 1.00 2433 2253
## StageWithoutseedinflow:Shannon 25.70 1.93 21.93 29.44 1.00 1792 1873
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 23.18 0.59 22.08 24.40 1.00 2808 2294
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.296204 0.02350415 0.2492173 0.3402776
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 83.12 25.51 33.18 134.09 1.00 4919 2944
## Ninitial4:StageWithseedinflow 151.76 24.44 105.83 199.35 1.00 5105 2673
## Ninitial8:StageWithseedinflow 166.70 19.10 130.40 203.15 1.00 4500 2749
## Ninitial16:StageWithseedinflow 250.46 17.52 216.04 284.94 1.00 4987 2903
## Ninitial32:StageWithseedinflow 278.91 19.71 239.91 317.22 1.00 4908 2609
## Ninitial2:StageWithoutseedinflow 23.07 10.19 3.56 43.26 1.00 5483 3074
## Ninitial4:StageWithoutseedinflow 64.69 14.55 36.70 92.63 1.00 5670 2851
## Ninitial8:StageWithoutseedinflow 89.13 14.22 61.36 117.73 1.00 4971 2888
## Ninitial16:StageWithoutseedinflow 184.81 17.91 149.85 219.44 1.00 4928 3186
## Ninitial32:StageWithoutseedinflow 189.47 23.32 142.71 235.02 1.00 6235 2422
## Ninitial2:StageWithseedinflow:Shannon -36.70 15.88 -68.43 -5.39 1.00 4912 2900
## Ninitial4:StageWithseedinflow:Shannon -54.37 11.18 -76.18 -33.14 1.00 5154 2872
## Ninitial8:StageWithseedinflow:Shannon -47.49 7.19 -61.33 -33.68 1.00 4293 2852
## Ninitial16:StageWithseedinflow:Shannon -64.87 5.96 -76.64 -53.25 1.00 4984 2807
## Ninitial32:StageWithseedinflow:Shannon -64.88 6.23 -76.98 -52.60 1.00 4930 2644
## Ninitial2:StageWithoutseedinflow:Shannon -3.36 6.83 -16.61 9.79 1.00 5501 3010
## Ninitial4:StageWithoutseedinflow:Shannon -19.36 7.47 -33.62 -5.04 1.00 5613 2798
## Ninitial8:StageWithoutseedinflow:Shannon -23.93 6.33 -36.57 -11.55 1.00 4946 2917
## Ninitial16:StageWithoutseedinflow:Shannon -52.76 7.38 -67.14 -38.46 1.00 4975 3208
## Ninitial32:StageWithoutseedinflow:Shannon -44.28 8.90 -62.12 -26.38 1.00 6256 2552
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 15.04 0.44 14.21 15.95 1.00 7394 3056
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.6994955 0.01128394 0.6755486 0.7195096
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 44.52 1.61 41.36 47.70 1.00 2125 2101
## StageWithoutseedinflow 37.52 1.79 33.95 41.11 1.00 2081 2146
## StageWithseedinflow:Shannon 8.71 0.56 7.57 9.80 1.00 2156 2175
## StageWithoutseedinflow:Shannon 15.08 0.79 13.49 16.64 1.00 1978 2150
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 12.22 0.31 11.60 12.84 1.00 2951 2255
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4415411 0.01983217 0.4003317 0.4789923
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 84.84 33.26 20.52 149.85 1.00 5106 2557
## Ninitial4:StageWithseedinflow 135.66 35.43 67.21 206.42 1.00 4361 2258
## Ninitial8:StageWithseedinflow 133.36 31.45 73.46 194.25 1.00 4089 2679
## Ninitial16:StageWithseedinflow 94.00 49.50 -1.22 188.99 1.00 4620 2745
## Ninitial32:StageWithseedinflow 41.88 60.77 -78.97 159.87 1.00 4763 2773
## Ninitial2:StageWithoutseedinflow -16.60 23.17 -60.09 29.90 1.00 4681 2505
## Ninitial4:StageWithoutseedinflow 27.33 13.80 -0.08 54.41 1.00 5324 2756
## Ninitial8:StageWithoutseedinflow 38.95 10.86 17.72 60.86 1.00 4866 2693
## Ninitial16:StageWithoutseedinflow 47.75 11.36 25.45 69.78 1.00 4839 2670
## Ninitial32:StageWithoutseedinflow 72.53 14.90 43.30 101.07 1.00 4600 2721
## Ninitial2:StageWithseedinflow:Shannon -16.05 20.16 -55.34 22.78 1.00 5101 2574
## Ninitial4:StageWithseedinflow:Shannon -29.65 15.35 -60.30 0.14 1.00 4384 2230
## Ninitial8:StageWithseedinflow:Shannon -20.19 10.86 -41.24 0.50 1.00 4100 2854
## Ninitial16:StageWithseedinflow:Shannon -5.12 13.97 -32.00 21.75 1.00 4601 2885
## Ninitial32:StageWithseedinflow:Shannon 8.38 14.48 -19.83 37.18 1.00 4825 2774
## Ninitial2:StageWithoutseedinflow:Shannon 46.15 14.33 17.11 73.36 1.00 4697 2508
## Ninitial4:StageWithoutseedinflow:Shannon 22.04 7.50 7.37 36.94 1.00 5408 2919
## Ninitial8:StageWithoutseedinflow:Shannon 17.58 4.91 7.72 27.16 1.00 4887 2792
## Ninitial16:StageWithoutseedinflow:Shannon 12.12 4.16 4.10 20.37 1.00 4820 2662
## Ninitial32:StageWithoutseedinflow:Shannon 3.27 4.45 -5.22 12.02 1.00 4557 2702
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.04 0.25 8.57 9.54 1.00 5731 2754
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4943489 0.01956404 0.4532455 0.5312383
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 46.06 1.40 43.31 48.82 1.00 1952 2034
## StageWithoutseedinflow 46.05 1.47 43.25 49.02 1.00 2143 2182
## StageWithseedinflow:Shannon 2.70 0.52 1.68 3.70 1.00 1945 2316
## StageWithoutseedinflow:Shannon 2.87 0.59 1.65 3.98 1.00 2174 2135
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.96 0.26 9.47 10.49 1.00 3156 2237
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.06594951 0.01680781 0.03504214 0.1020468
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 18.47 37.70 -56.05 91.52 1.00 3683 2748
## Ninitial4:StageWithseedinflow -7.56 25.81 -57.57 43.36 1.00 3347 2581
## Ninitial8:StageWithseedinflow 3.49 25.38 -46.93 52.91 1.00 3560 2610
## Ninitial16:StageWithseedinflow 58.70 23.40 13.04 106.65 1.00 3809 2744
## Ninitial32:StageWithseedinflow 48.44 30.05 -10.79 105.72 1.00 3673 2738
## Ninitial2:StageWithoutseedinflow 64.80 6.57 51.70 77.79 1.00 3516 2795
## Ninitial4:StageWithoutseedinflow 25.63 11.17 4.53 47.52 1.00 3151 2517
## Ninitial8:StageWithoutseedinflow 37.11 11.78 14.09 60.26 1.00 3568 2570
## Ninitial16:StageWithoutseedinflow 65.50 21.54 23.32 107.83 1.00 3754 2854
## Ninitial32:StageWithoutseedinflow 79.00 33.06 15.36 144.51 1.00 3660 2525
## Ninitial2:StageWithseedinflow:Shannon 18.26 22.59 -25.79 62.96 1.00 3682 2746
## Ninitial4:StageWithseedinflow:Shannon 25.33 11.31 2.89 47.30 1.00 3349 2616
## Ninitial8:StageWithseedinflow:Shannon 17.57 8.97 -0.04 35.41 1.00 3542 2506
## Ninitial16:StageWithseedinflow:Shannon -1.02 6.94 -15.25 12.66 1.00 3809 2724
## Ninitial32:StageWithseedinflow:Shannon 2.46 7.90 -12.69 17.98 1.00 3655 2823
## Ninitial2:StageWithoutseedinflow:Shannon -9.94 4.07 -18.06 -1.85 1.00 3515 2707
## Ninitial4:StageWithoutseedinflow:Shannon 11.53 5.22 1.30 21.34 1.00 3167 2553
## Ninitial8:StageWithoutseedinflow:Shannon 6.08 4.49 -2.76 14.85 1.00 3589 2544
## Ninitial16:StageWithoutseedinflow:Shannon -3.60 6.77 -16.87 9.67 1.00 3732 2860
## Ninitial32:StageWithoutseedinflow:Shannon -5.79 9.40 -24.41 12.40 1.00 3657 2525
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 6.73 0.19 6.37 7.11 1.00 5563 2621
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2345123 0.02483178 0.1862827 0.2825331
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 31.35 2.03 27.44 35.31 1.00 2048 2039
## StageWithoutseedinflow 27.91 2.25 23.51 32.33 1.00 2161 2262
## StageWithseedinflow:Shannon 3.76 1.21 1.42 6.09 1.00 2131 2002
## StageWithoutseedinflow:Shannon 3.81 1.66 0.63 7.06 1.00 2165 2334
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 13.36 0.34 12.73 14.08 1.00 2845 2437
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.04752129 0.01399466 0.02318427 0.07695434
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 43.41 7.61 28.58 58.21 1.00 4339 3259
## Ninitial4:StageWithseedinflow 49.09 6.37 36.67 61.51 1.00 4208 2878
## Ninitial8:StageWithseedinflow 47.12 7.41 32.95 61.80 1.00 4085 3033
## Ninitial16:StageWithseedinflow 36.67 9.41 18.63 55.35 1.00 4348 3192
## Ninitial32:StageWithseedinflow 17.89 14.05 -9.71 45.79 1.00 4288 3195
## Ninitial2:StageWithoutseedinflow 45.29 9.53 26.50 64.19 1.00 3924 3113
## Ninitial4:StageWithoutseedinflow 42.32 6.07 30.55 54.00 1.00 3769 2662
## Ninitial8:StageWithoutseedinflow 52.11 6.75 38.58 65.01 1.00 4158 2544
## Ninitial16:StageWithoutseedinflow 18.27 6.43 5.88 31.12 1.00 4692 2921
## Ninitial32:StageWithoutseedinflow 10.35 5.88 -1.24 21.77 1.00 4147 2893
## Ninitial2:StageWithseedinflow:Shannon -7.99 6.35 -20.51 4.59 1.00 4315 2984
## Ninitial4:StageWithseedinflow:Shannon -8.62 4.76 -17.96 0.59 1.00 4284 2986
## Ninitial8:StageWithseedinflow:Shannon -4.51 4.79 -13.84 4.45 1.00 4147 3079
## Ninitial16:StageWithseedinflow:Shannon 1.97 4.75 -7.37 11.08 1.00 4344 2965
## Ninitial32:StageWithseedinflow:Shannon 8.92 5.70 -2.40 20.21 1.00 4329 3154
## Ninitial2:StageWithoutseedinflow:Shannon -13.79 8.93 -31.71 3.83 1.00 3908 3059
## Ninitial4:StageWithoutseedinflow:Shannon -7.38 4.94 -16.91 2.27 1.00 3652 2962
## Ninitial8:StageWithoutseedinflow:Shannon -12.28 5.03 -21.98 -2.17 1.00 4150 2767
## Ninitial16:StageWithoutseedinflow:Shannon 10.99 4.11 2.58 18.86 1.00 4802 2988
## Ninitial32:StageWithoutseedinflow:Shannon 12.64 3.27 6.38 18.97 1.00 3871 2812
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 11.84 0.34 11.22 12.53 1.00 7498 2740
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1468626 0.0223651 0.1037254 0.1903548
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 91.66 0.49 90.70 92.62 1.00 1847 1983
## StageWithoutseedinflow 71.71 0.56 70.62 72.83 1.00 1921 1977
## StageWithseedinflow:Shannon 0.55 0.18 0.20 0.91 1.00 1858 1742
## StageWithoutseedinflow:Shannon 0.19 0.30 -0.43 0.78 1.00 1795 2165
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 3.56 0.09 3.38 3.74 1.00 3153 2601
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.8971751 0.002213982 0.8923178 0.9010684
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 98.48 3.71 91.40 105.70 1.00 3459 2909
## Ninitial4:StageWithseedinflow 99.78 4.37 91.19 108.39 1.00 3693 2815
## Ninitial8:StageWithseedinflow 94.85 6.35 82.42 107.51 1.00 3396 2718
## Ninitial16:StageWithseedinflow 96.22 10.52 75.07 116.94 1.00 3823 2796
## Ninitial32:StageWithseedinflow 93.11 15.33 62.48 123.41 1.00 3783 2912
## Ninitial2:StageWithoutseedinflow 67.57 2.07 63.48 71.54 1.00 4180 3007
## Ninitial4:StageWithoutseedinflow 68.60 1.96 64.57 72.43 1.00 3329 3059
## Ninitial8:StageWithoutseedinflow 73.93 2.14 69.75 78.02 1.00 3367 2993
## Ninitial16:StageWithoutseedinflow 66.25 2.02 62.23 70.18 1.00 3875 3274
## Ninitial32:StageWithoutseedinflow 71.19 2.69 65.84 76.41 1.00 3790 3083
## Ninitial2:StageWithseedinflow:Shannon -4.02 2.43 -8.81 0.59 1.00 3420 2828
## Ninitial4:StageWithseedinflow:Shannon -2.74 2.06 -6.80 1.29 1.00 3673 2754
## Ninitial8:StageWithseedinflow:Shannon -0.34 2.37 -5.06 4.29 1.00 3399 2709
## Ninitial16:StageWithseedinflow:Shannon -0.85 3.14 -7.09 5.44 1.00 3821 2947
## Ninitial32:StageWithseedinflow:Shannon 0.02 3.90 -7.66 7.74 1.00 3778 3030
## Ninitial2:StageWithoutseedinflow:Shannon 3.85 1.53 0.88 6.83 1.00 4228 2889
## Ninitial4:StageWithoutseedinflow:Shannon 3.04 1.23 0.67 5.50 1.00 3340 2885
## Ninitial8:StageWithoutseedinflow:Shannon -0.76 1.07 -2.84 1.32 1.00 3439 2781
## Ninitial16:StageWithoutseedinflow:Shannon 2.68 0.95 0.85 4.59 1.00 3884 3221
## Ninitial32:StageWithoutseedinflow:Shannon -0.09 1.09 -2.22 2.08 1.00 3786 2964
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 2.97 0.09 2.81 3.15 1.00 7693 2799
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.9274615 0.001592317 0.9240772 0.9302001
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs acrosstain species diversity in a process-based model of succulent plant communities. Ecological Modelling.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 63.05 1.51 60.19 66.16 1.00 1829 2059
## StageWithoutseedinflow 43.49 2.03 39.62 47.61 1.01 1809 1841
## StageWithseedinflow:Shannon 2.02 0.64 0.71 3.22 1.00 1858 1850
## StageWithoutseedinflow:Shannon 4.55 1.28 2.01 7.01 1.01 1745 1852
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 10.80 0.29 10.28 11.40 1.00 2810 2178
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.3977007 0.02144508 0.3557756 0.4375859
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow 101.29 6.40 88.85 113.89 1.00 3423 2588
## Ninitial4:StageWithseedinflow 95.54 8.95 78.23 113.33 1.00 4234 2815
## Ninitial8:StageWithseedinflow 105.48 11.14 83.62 127.02 1.00 3656 2891
## Ninitial16:StageWithseedinflow 103.41 15.96 72.09 134.72 1.00 3677 3089
## Ninitial32:StageWithseedinflow 131.68 20.32 92.09 171.10 1.00 3509 3222
## Ninitial2:StageWithoutseedinflow 50.37 4.29 41.93 58.93 1.00 3758 2898
## Ninitial4:StageWithoutseedinflow 66.04 4.53 57.07 75.00 1.00 3849 2941
## Ninitial8:StageWithoutseedinflow 63.88 4.69 54.52 73.23 1.00 4032 3020
## Ninitial16:StageWithoutseedinflow 79.92 6.62 67.00 93.15 1.00 4063 2945
## Ninitial32:StageWithoutseedinflow 77.72 9.58 59.69 97.23 1.00 3830 2898
## Ninitial2:StageWithseedinflow:Shannon -24.71 4.41 -33.44 -15.98 1.00 3578 2634
## Ninitial4:StageWithseedinflow:Shannon -14.05 4.69 -23.34 -4.95 1.00 4222 2430
## Ninitial8:StageWithseedinflow:Shannon -15.35 4.58 -24.21 -6.30 1.00 3628 2871
## Ninitial16:StageWithseedinflow:Shannon -11.46 5.40 -22.05 -0.91 1.00 3678 3110
## Ninitial32:StageWithseedinflow:Shannon -17.75 5.87 -29.22 -6.29 1.00 3519 3197
## Ninitial2:StageWithoutseedinflow:Shannon -3.37 3.37 -10.05 3.20 1.00 3823 2987
## Ninitial4:StageWithoutseedinflow:Shannon -10.29 3.13 -16.38 -4.10 1.00 3807 2880
## Ninitial8:StageWithoutseedinflow:Shannon -7.05 2.79 -12.50 -1.60 1.00 4062 3171
## Ninitial16:StageWithoutseedinflow:Shannon -13.88 3.56 -20.92 -6.83 1.00 4111 3090
## Ninitial32:StageWithoutseedinflow:Shannon -11.14 4.77 -20.91 -2.10 1.00 3828 2731
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 7.97 0.22 7.56 8.42 1.00 6281 2556
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5794212 0.0160797 0.5456339 0.608345
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.